
TL;DR
This paper explores fundamental questions in the mathematical theory of domains used in denotational semantics, introducing new approaches and clearer justifications for understanding subdomains, retractions, generalizations, and metrics.
Contribution
It presents novel methods and comprehensive answers to key questions about domain structures, subdomains, retractions, and metrics, advancing the theoretical understanding of domains in semantics.
Findings
Subdomains of a domain form a domain under certain conditions.
Finitary retractions are characterized within the domain framework.
Relaxed metrics can be obtained via co-continuous valuations.
Abstract
Two groups of naturally arising questions in the mathematical theory of domains for denotational semantics are addressed. Domains are equipped with Scott topology and represent data types. Scott continuous functions represent computable functions and form the most popular continuous model of computations. Covariant Logic of Domains: Domains are represented as sets of theories, and Scott continuous functions are represented as input-output inference engines. The questions addressed are: A. What constitutes a subdomain? Do subdomains of a given domain form a domain? B. Which retractions are finitary? C. What is the essence of generalizations of information systems based on non-reflexive logics? Are these generalizations restricted to continuous domains? Analysis on Domains: D. How to describe Scott topologies via generalized distance functions satisfying the requirement of Scott…
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Taxonomy
TopicsSemantic Web and Ontologies · Rough Sets and Fuzzy Logic · Logic, Reasoning, and Knowledge
